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Estimating stable fixed points and Langevin potentials for financial dynamics
Tobias Wand1, Timo Wiedemann2, Jan Harren2
1Institut für Theoretische Physik, Universität Münster, Wilhelm-Klemm-Str. 9, 48149 Münster, Germany and Center for Nonlinear Science, Universität Münster, Corrensstr. 2, 48149 Münster, Germany.
Abstract:
The geometric Brownian motion (GBM) is a standard model in quantitative finance, but the potential function of its stochastic differential equation (SDE) cannot include stable nonzero prices. This article generalizes the GBM to an SDE with polynomial drift of order q and shows via model selection that q=2 is most frequently the optimal model to describe the data. Moreover, Markov chain Monte Carlo ensembles of the accompanying potential functions show a clear and pronounced potential well, indicating the existence of a stable price.
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